Quentin Anciaux skrev:
> Le Thursday 29 November 2007 17:22:59 Torgny Tholerus, vous avez écrit :
>
>>
>> There is a difference between "unlimited" and "infinite". "Unlimited"
>> just says that it has no limit, but everything is still finite. If you
>> add something to a finite set, then the new set will always be finite.
>> It is not possible to create an infinite set.
>>
>
> I'm sorry I don't get it... The set N as an infinite numbers of elements still
> every element in the set is finite. Maybe it is an english subtility that I'm
> not aware of... but in french I don't see a clear difference between "infini"
> and "illimité".
>
As soon as you talk about "the set N", then you are making a "closure"
and making that set finite. The only possible way to talk about
something without limit, such as natural numbers, is to give a
"production rule", so that you can produce as many of that type of
objects as you want. If you have a natural number n, then you can
"produce" a new number n+1, that is the successor of n.
>
>
>> So it is OK to use the word "unlimited". But it is not OK to use the
>> word "infinite". Is this clear?
>>
>
> No, I don't see how a set which have not limit get a finite number of
> elements.
>
It is not possible for "a set" to have no limit. As soon as you
construct "a set", then that set will always have a limit. Either you
have to accept that the set N is finite, or you must stop talking about
"the set N". It is enough to have a production rule for natural numbers.
>
>> Another important word is the word "all". You can talk about "all
>> events". But in that case the number of events will be finite, and you
>> can then talk about "the last event". But you can't deduce any
>> contradiction from that, because that is forbidden by the type theory.
>> And there will be more events after "the last event", because the number
>> of events is "unlimited".
>>
>
> If there are events after the last one, how can the last one be the last ?
>
The last event is the last event in the set of "all" events. But
because you have a production rule for the events, it is always possible
to produce new events after the last event. But these events do not
belong to the set of "all" events.
>
>> As soon as you use the word "all", you will
>> introduce a limit - all up to this limit. And you must then think of
>> only doing conclusions that are legal according to type theory.
>>
>
> o_O... could you explain what is type theory ?
>
Type theory is one of the solutions of Russel's paradox. You have a
hierarchy of "types". Type theory says that the "all quantifiers" only
can span objects of the same "type" (or lower types). When you create
new objects, such that "the set of all sets that do not belong to
themselves", then you will get an object of a higher "type", so that you
can not say anything about if this set belongs to itself or not. The
same thing with "the set of all sets". You can not say anything about
if it belongs to itself.
>
>> So the best thing is to avoid the word "all" (and all synonyms of that
>> word).
>>
>
> like everything ?
>
Yes... :-)
--
Torgny
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Received on Thu Nov 29 2007 - 12:35:34 PST